doubt

Respected sir,
I hava the following doubt. please clarify my doubt.

let x belongs to {1}.
<=> x belongs to any one of the subsets of {1}.
<=> x=1 or x belongs to {}.
But generally every body will write
x=1 <=> x belongs to {1}.
But for to use <=> we should write:
x belongs to{1} and x does not belongs to {}.
Why dont every one write like this.

Because this statement is not true. has only one element. That is to say that . is indeed a subset of , but no element of this set is equal to because the empty set is not listed as one of the set members.

let x belongs to {1}.
<=> x belongs to any one of the subsets of {1}.
<=> x=1 or x belongs to {}.
But generally every body will write
x=1 <=> x belongs to {1}.
But for to use <=> we should write:
x belongs to{1} and x does not belongs to {}.
Why dont every one write like this.